The reaction rates of the beta processes for all particles at arbitrary degeneracy are derived, and an analytic steady state equilibrium condition μn=μp+2μe which results from the equality of electron and positron capture rates in the hot electron-positron plasma with nucleons is also found, if the matter is transparent to neutrinos. This simple analytic formula is valid only if electrons are nondegenerate or mildly degenerate, which is generally satisfied in the hot electron-positron plasma. Therefore, it can be used to efficiently determine the steady state of the hot matter with plenty of positrons. Based on this analytic condition, given the baryon number density and the temperature, if the nucleons are nondegenerate, only one algebraic equation for determining the electron fraction is obtained, which shows the great advantage of the analytic equilibrium condition.

• Received 12 November 2004

DOI:https://doi.org/10.1103/PhysRevD.72.013007